AbstractThe Lagrangian of the general‐relativistic affine field theory of the non‐symmetric connection field Γiklis Schrödinger scalar density\documentclass{article}\pagestyle{empty}\begin{document}$ H = \frac{2}{\lambda }\sqrt { - \det [R} _{ik} ] $\end{document}, and the field variables (canonical coordinates) are Einstein's affine tensorsUmnl= Γlmn‐ δnlΓrmn. The field equations are the Einstein‐Schrödinger equations\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{\delta H}}{{\delta U^i _{mn} }} = N^{mn} _i = 0 $$\end{document}The minors\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{\partial H}}{{\partial R_{mn} }} = G^{mn} = \sqrt { - gg^{mn} } $$\end{document}give by definitongmn= λ−1Rmn, and λ becomes the cosmological constant. The Hamiltonian density is theV00‐component of the Einstein energy‐momentum complex\documentclass{article}\pagestyle{empty}\begin{document}$$ V_i ^k = \frac{{\partial H}}{{\partial U^i _{mn,k} }}U^i _{mn,i} = - \frac{1}{{2\lambda ^2 }}HR^{mn} \partial _i^k U^i _{mn.i} $$\end{document}and the tensor‐density components\documentclass{article}\pagestyle{empty}\begin{document}$$ \gamma ^{mn} _i = - \frac{1}{{2\lambda ^2 }}HR^{mn} \partial _i^0 = - \sqrt { - gg^{mn} } \partial _i^0 = g^{mn} \partial _1^0 $$\end{document}are the canonically conjugated momentum densities of the field coordinatesUlmn. The canonical equations are\documentclass{article}\pagestyle{empty}\begin{document}$$ ( - g)^{ - \frac{1}{2}} N^{mn} _{iV_0^0 } = 0 $$\end{document}, and we have no constraints. The affine field theory is invariant with respect to all transformations which preserve the Levi‐Civita parallelism (Einstein's unifiedT‐Agroup), and the field equations possess transposition invariance: WithŨlmn=Ulnmwe getR̃mn=Rnm,g̃mn=gnm, andÑmnl=Unml.The symmetry conditions Γimn= Γinmreduce the space to the general‐relativistic Einstein spaces withRik=Rki. The equationRik= λgikyields Γikl= {ikl}, and the pathes of test particles define geodes